Shear Performance and Load–Slip Model of a Cross-Type FRP Rod Connector for Precast Concrete Sandwich Panels

Abstract

A precast concrete sandwich panel (PCSP), consisting of inner and outer wythes, an insulation layer, and connectors, relies heavily on the shear behavior of these connectors, which governs the structural performance of the entire system. Owing to their low thermal conductivity, excellent durability, and high strength, fiber-reinforced polymer (FRP) connectors offer strong potential for widespread application. This study introduces a novel cross-shaped FRP rod connector designed to provide improved anchorage performance, bidirectional shear resistance, and ease of installation. However, concern remains about the specific influence of embedment depth, outer-wythe thickness, and insulation-layer thickness on its shear performance. Moreover, no calculation model for shear capacity or shear–slip model has been established considering the shear-bending interaction within the connector. To evaluate its shear behavior, six groups of push-out tests were conducted, with key parameters including embedment depth, outer-wythe thickness, and insulation-layer thickness. The specimens exhibited two primary failure modes: connector fracture and concrete anchorage failure. The measured shear capacity per connector ranged from 5.63 kN to 14.19 kN, increasing with longer embedment depths, decreasing with increasing insulation thickness, and showing no clear dependence on outer-wythe thickness. Guided by test results and the Hashin failure criterion for composite materials, analytical formulas to estimate the shear capacity of FRP connectors were developed. The mean ratio of calculated to experimental values is 0.97, with a standard deviation of 0.06, indicating good agreement between the predicted and measured shear capacities. Furthermore, a theoretical shear–slip model was established. The correlation coefficients between the experimental and calculated load–slip curves for all specimens are greater than 0.98, indicating a high consistency in curve shape and variation trend.

1. Introduction

A typical precast concrete sandwich panel (PCSP) consists of inner and outer concrete wythes, an insulation layer, and connectors, integrating structural load-bearing capacity with thermal insulation and architectural functionality [1]. Owing to these advantages, PCSPs are widely employed as exterior enclosure walls in both public and residential buildings. PCSPs are primarily designed to resist out-of-plane loads, including wind pressures and seismic actions. Under such loading conditions, shear forces developed between the wythes are transferred predominantly via the connectors. Consequently, the shear performance of these connectors plays a pivotal role in the structural safety and overall performance of PCSPs.

Early applications of PCSPs commonly utilized solid concrete zone between the two wythes as connectors, which provided high stiffness and strong load-bearing capacity [2]. However, the concrete penetrating the insulation layer creates thermal bridges, significantly reducing the thermal insulation performance of the wall system. By the 1980s, steel connectors became the predominant choice for insulated sandwich walls [2,3,4,5]. Nevertheless, on account of the elevated thermal conductivity of steel, thermal bridges at the connector locations remained a persistent issue, making it difficult to meet increasingly stringent energy efficiency standards. Additionally, the limited corrosion resistance of steel connectors could give rise to long-term durability concerns. Over the following decade, fiber-reinforced polymer (FRP) connectors gained widespread adoption in PCSPs owing to their superior thermal efficiency [4]. Characterized by low thermal conductivity, excellent durability, and high strength, FRP connectors effectively mitigate thermal bridging at the connections, thereby enhancing both the energy efficiency and long-term safety of the wall system. These advantages make FRP connectors a highly promising solution for broad application in the construction industry.

The existing literature provides substantial experimental data on the shear performance of FRP connectors. Einea [4], Natio [6], Woltman [7], Hodicky [2], Huang [8], Chen [9], and Liu [10] performed shear tests on different types of FRP connectors and compared their performance with that of the steel connectors. These studies indicated that the shear strength of FRP connectors was lower than that of both steel connectors and the pure shear strength of the FRP material itself, while their shear stiffness was also inferior to that of steel connectors. This phenomenon occurs due to connectors in PCSPs experience a combination of shear forces, bending moments, and axial forces arise from the slip between the two wythes. As a result, their ultimate load-carrying capacity cannot be directly predicted using the pure shear strength of FRP materials. Moreover, because FRP materials lack a yield plateau, the connectors are unable to undergo internal force redistribution, further limiting their shear performance. Natio [6] investigated the shear behavior of FRP connectors with various geometric configurations. The results showed that FRP rod connectors exhibited shear capacities ranging from 1.5 kN to 8.4 kN, whereas plate connectors achieved a capacity of 11.9 kN, and grid connectors reached 40.3 kN/m. The study further indicated that connector geometry has a pronounced influence on shear stiffness, with plate connectors and grid connectors demonstrating substantially higher stiffness than rod connectors. Meng et al. [11] conducted shear tests on U-shaped and V-shaped FRP connectors. Their results showed that U-shaped connectors exhibited higher shear strength, whereas V-shaped connectors demonstrated smaller ultimate displacement. Furthermore, both the embedment depth and the insulation layer thickness significantly influence the shear performance of the connector. An increase in insulation layer thickness leads to a reduction in the shear capacity of the connector and shifts the failure mode from connector fracture to concrete anchorage failure [12,13,14,15]. Conversely, an increase in the embedment depth enhances the shear capacity [16,17]. As demonstrated in Choi’s study, a truss-type GFRP connector with an embedment depth of 30 mm failed due to anchorage failure, whereas a connector with a 40 mm embedment depth failed by fracture [16]. Generally, increasing the orientation angle of connectors also increased their stiffness and ultimate capacity, while the most effective orientation angle observed by Fahmy et al. was 45 degrees from tested specimens [18]. However, it is difficult to precisely control this installation angle during construction. Furthermore, the out-of-plane shear performance of connectors at such an angle has not been validated.

With regard to theoretical studies on the shear behavior of connectors, prior research has primarily focused on developing computational models for shear capacity and shear–slip relationships. Woltman et al. presented a calculation model for the shear resistance of FRP rod connectors [7]. This model analyzed the bending and shear actions separately without considering the material failure criteria under their combined effect, leading to an overestimation of the connector’s shear capacity. Bunn [19] suggested that the shear capacity of connectors is influenced by factors such as the type of insulation layer, insulation thickness, connector spacing, and orientation. He introduced a calculation formula for the shear capacity of CFRP grid connectors, in which various correction coefficients were fitted based on limited experimental data, thus lacking general applicability. Liu et al. proposed a model for shear capacity of BFRP bar connectors based on a quartic polynomial response surface model and cross-validation error analysis method [10]. This model fits the shear capacity of the connectors using the angle and span-to-height ratio as variables; however, it lacks a solid theoretical foundation. Based on shear tests of 14 types of connectors, Natio et al. [6] developed a tri-linear shear–slip model consisting of elastic, plastic, and unloading phases. In this model, the plastic phase initiates when the shear force reaches 75% of the connector’s shear capacity, and the ultimate slip is taken as the slip at which the shear force has dropped by 50%. Nonetheless, no analytical approach was provided to determine the characteristic slip points required for practical use of the model. In practice, the slip between concrete wythes can be decomposed into two components: the slip at the anchored end of the connector, and the slip due to connector deformation within the height of the insulation layer. To date, no study has separately considered these two slip components to propose a comprehensive calculation model for the whole shear–slip response of FRP connectors.

The International Code Council Evaluation Service (ICC-ES), an internationally recognized professional evaluation agency based in the United States, published the AC320 in 2015 [20]. This standard specifies requirements for testing methods, performance evaluation, quality control, and certification for FRP connectors. Subsequently, the Chinese standard JG/T 561-2019 [21] stipulates the material properties, mechanical performance, durability, pull-out and shear performance requirements, and corresponding test methods for FRP connectors. However, current standards provide neither a unified calculation method for the shear capacity nor a definitive shear–slip model for connectors.

In summary, FRP connectors are capable of meeting the design requirements of PCSPs. Experimental studies on connectors with various geometric configurations have shown that geometry significantly affects shear behavior: plate connectors and grid connectors exhibit relatively high load-bearing capacity and stiffness under in-plane shear, whereas their out-of-plane shear performance is considerably lower. In contrast, rod connectors provide bidirectional shear resistance. With reasonable structural design to improve their shear capacity, they can serve as excellent insulated connectors. Furthermore, the shear performance of connectors is also affected by factors such as the embedment depth and insulation thickness; however, the specific influence of these factors on shear performance still remains insufficiently understood. In terms of theoretical research, no calculation model for shear capacity or shear–slip model has been established considering the shear-bending interaction within the connector. To address the above issues, this study develops a novel FRP rod connector with a cruciform cross-section. An experimental program of shear tests was carried out to investigate the effects of embedment depth, outer-wythe thickness, and insulation thickness on connector’s shear capacity and shear–slip response. Based on the test results and the Hashin criterion for composite materials, a calculation method for shear capacity and a shear–slip model for the proposed connector are established.

2. Connector Design and Development

Beginning in 2007, our team has developed a cross-shaped FRP rod connector to meet the connection needs of PCSP. The connector comprises two primary components: an FRP core and a plastic collar. The FRP core is designed to transfer inter-wythe shear, while the plastic collar ensures accurate positioning within the insulation layer and fixity during concrete casting. For improved anchorage, triangular grooves are cut into the concrete-embedded portions at the ends of each flange of the FRP core (see Figure 1). The FRP core was made of unidirectional continuous TM-glass fibers impregnated in a vinyl ester resin matrix using the pultrusion process. Compared to conventional E-glass fiber, TM-glass fiber has higher tensile strength, tensile modulus, alkaline resistance and acid resistance. The volume fractions of fibers, resin, and other ingredients are presented in

Table 1. Volume fraction of each ingredient in the FRP core.

Glass Fiber	Vinyl Ester Resin	Hardeners and Mold Release Agent
75%	22%	3%

. Subsequently, triangular grooves were created on both ends of each flange within the region to be anchored in concrete. Finally, a plastic collar was formed around the exterior of the FRP core via injection molding. The material of the collar is a copolymer of acrylonitrile, butadiene, and styrene (ABS), which exhibits excellent thermal efficiency, heat resistance, low-temperature resistance, chemical resistance, impact resistance, and electrical properties, along with good processability and cost-effectiveness.

To evaluate the mechanical properties of the connector, tensile tests and short-beam shear tests were conducted. The tensile tests were performed in accordance with the Chinese national standard GB/T 1447-2005 [22], while the short-beam shear tests followed ASTM D2344-22 [23]. The measured material properties are summarized in

Table 2. Mechanical properties of the connector.

Tensile Strength (N/mm2)	Tensile Modulus (×103 N/mm2)	Short-Beam Shear Strength (N/mm2)
774.1	46.27	48.44

.

3. Experimental Program

3.1. Test Specimen

With reference to Eurocode 4 [24], a total of six groups comprising 18 samples were prepared for the push-out test. The test variables included: (i) the embedment depth of the connectors (30 mm and 50 mm); (ii) the thickness of the outer wythe (60 mm, 150 mm, and 200 mm); and (iii) the thickness of the insulation layer (30 mm, 70 mm, 90 mm, and 120 mm).

Each specimen consisted of three concrete wythes connected by four connectors, with two connectors placed between each pair of adjacent wythes. The spacing between connectors was 500 mm. Each concrete wythe was 1000 mm long by 500 mm wide. The thickness of the two outer wythes was selected to represent practical dimensions commonly adopted in engineering applications, while the thickness of the inner wythe was twice as thick as the outer wythe. The specimen designations and corresponding parameters are listed in

Table 3. Test matrix.

Specimen No.	Embedment Depth of the Connector (mm)	Outer Wythe Thickness (mm)	Insulation Layer Thickness (mm)
PS-50-150-30-1~3	50	150	30
PS-75-150-30-1~3	75	150	30
PS-75-200-30-1~3	75	200	30
PS-30-60-70-1~3	30	60	70
PS-30-60-90-1~3	30	60	90
PS-30-60-120-1~3	30	60	120

, and the specimen configuration is illustrated in Figure 2.

3.2. Material

All specimens were prepared using a concrete mixture designed for a cubic compressive strength of 30 MPa, and steel reinforcements with a characteristic yield strength of 400 MPa.

Table 4. Mechanical properties of steel reinforcements.

Nominal Diameter (mm)	Yield Strength (N/mm2)	Ultimate Strength (N/mm2)	Elastic Modulus (×105 N/mm2)	Elongation (%)
8	497	682	2.21	21.2

and

Table 5. Mechanical properties of concrete.

Specimen No.	Cubic Compressive Strength fcu (N/mm2)	Prismatic Compressive Strength fc (N/mm2)	Split Strength ft (N/mm2)	Elastic Modulus Ec (×104 N/mm2)
PS-50-150-30-1~3	38.4	28.5	2.83	3.22
PS-75-150-30-1~3	39.2	27.8	2.69	3.07
PS-75-200-30-1~3	39.9	29.2	2.91	3.14
PS-30-60-70-1~3	35.1	28.3	2.40	3.15
PS-30-60-90-1~3	35.1	28.3	2.40	3.15
PS-30-60-120-1~3	36.6	27.4	2.74	3.02

show the measured mechanical properties of the concrete and steel reinforcement.

3.3. Construction Process

The construction of the specimens simulated the production process of precast sandwich insulation walls, as shown in Figure 3. The procedure was as follows: first, the longitudinal and transverse reinforcement for the outer concrete wythe was placed within the formwork, followed by casting the concrete to the specified wythe thickness. Subsequently, a layer of XPS insulation board was laid on the cured outer wythe. Preformed holes for the connectors were reserved in the XPS board, into which the connectors were inserted. Next, the reinforcement for the inner concrete wythe was placed on the XPS board, and concrete was cast to the required inner wythe thickness. This process was repeated for the upper portion of the wall, with another XPS board laid, connectors installed, and concrete cast to form the upper outer wythe.

3.4. Test Setup and Measurements

The test setup for the push-out test is shown in Figure 4. A preloading stage was conducted first, followed by slow loading until specimen failure, with a controlled rate of 1 kN per minute. Crack initiation and slip deformation were tracked on the concrete surface during loading. Note that to eliminate the effect of the insulation board, the XPS foam was removed prior to testing.

The push-out test on connectors primarily included the following measurements:

(1) Strain in connectors

For each specimen, two connectors on one side were selected for strain measurement. Strain gauges were attached to the top and bottom flanges of the cruciform cross-section within the cavity of the insulation layer, near the concrete interface. On each flange, one strain gauge was aligned along the axial direction of the connector, one perpendicular to the axial direction, and one at a 45° angle relative to the axial direction (see Figure 5). A total of 12 strain gauges were thus installed per specimen.

(2) Vertical Slip between Connectors and Concrete wythes

Four LVDTs were installed per specimen to monitor the inter-wythe vertical slip. For specimens with an insulation layer thickness exceeding 70 mm, two connectors were selected, and two additional LVDTs were set at both ends of these connectors to record the vertical slip at the anchored ends of connectors (Figure 6).

4. Experimental Results

4.1. Overall Responses and Failure Mode

As the applied load increased to about 50% of the peak load, fiber-tearing sounds were observed in the connectors, indicating the initiation of damage and accompanied by noticeable tilting. At later loading stages, significant relative displacement developed between the wythes. In some specimens, the connectors failed abruptly in shear. In others, splitting cracks first appeared in the concrete near the anchorage zone of the connectors and rapidly propagated to form a circumferential crack. This was followed by concrete anchorage failure, characterized by a conical pull-out mode with a failure surface inclined at approximately 30–35° relative to the concrete surface. In both failure modes, the specimens exhibited a relatively sudden loss of load-carrying capacity.

When the insulation layer thickness ranged from 30 mm to 90 mm, the predominant failure mode was connector fracture (Figure 7a). In contrast, 120 mm thick insulation specimens exhibited concrete anchorage failure (Figure 7b).

The connector strain increased nearly linearly up to specimen failure. The strain parallel to the connector axis was relatively large, whereas the strains perpendicular to the axis and at the 45° direction were considerably smaller. This indicates that the transverse shear forces acting on the connector generated substantial normal stresses within the member. In addition, the axial strains measured at the upper and lower sections were essentially symmetrical.

4.2. Load–Slip Curves and Shear Capacities

Push-out test results, in the form of load–slip curves (Figure 8 and Figure 9), characterize the full-range shear stiffness of the connectors. Specifically, Figure 8 presents the load–slip curves between the concrete wythes, providing an overall assessment of the shear transfer performance of connectors. Figure 9, on the other hand, shows the load–slip curves at the connector anchored ends, illustrating the anchorage behavior and interaction between the connectors and the surrounding concrete.

Based on the load–slip curves, the specimens with a 30 mm insulation layer exhibited a sharp reduction in stiffness after reaching a certain load level, followed by an extended plateau region. As slip continued to increase beyond this plateau, the load began to rise again until shear failure of the connectors. Because the shear stiffness within the plateau region is zero or even negative, the maximum load attained prior to entering this stage is defined as the characteristic load, which is subsequently used to determine the shear capacity per connector. The peak load (P_max), characteristic load (P_char), and the corresponding slip between the wythes for each specimen are summarized in

Table 6. Characteristic values of load and displacement.

Specimen No.	Failure mode	Peak load Pmax (kN)	Slip at peak load Δmax (mm)	Characteristic load Pchar (kN)	Characteristic slip Δchar (mm)	Shear Capacity per Connector Vu (kN)
PS-50-150-30-1~3	Connector fracture	52.25	20.99	47.72	2.51	11.93
PS-75-150-30-1~3		119.35	20.98	56.75	2.28	14.19
PS-75-200-30-1~3		85.22	18.97	54.05	2.58	13.51
PS-30-60-70-1~3		28.14	9.02	28.14	9.02	7.04
PS-30-60-90-1~3		26.29	6.03	26.29	6.03	6.57
PS-30-60-120-1~3	Concrete anchorage failure	22.52	16.35	22.52	16.35	5.63

, along with the average values for each specimen group.

From Figure 8 and Figure 9 and Table 6, the following observations can be made:

(1) At the onset of loading, specimens PS-50-150-30-1 to 3 exhibited an approximately linear load–slip response between the wythes, with only minor slip. After the applied load reached approximately two-thirds of Pmax, the stiffness decreased markedly, accompanied by longitudinal splitting cracks (delamination) developing along the connector. Following the appearance of these cracks, the load–slip curve exhibited a descending branch. Subsequently, the loading mechanism of the connector shifted to a combined tension-shear action dominated by tension. When the slip increased to around 15 mm, the load began to rise again until final failure occurred.

(2) Specimens PS-75-150-30-1 to 3 and PS-75-200-30-1 to 3 showed load-slip curves similar in form to those of PS-50-150-30-1~3. All exhibited relatively high initial stiffness, followed by a pronounced reduction in stiffness after the load reached approximately 0.5–0.9 Pmax, during which noticeable shear deformation of the connectors was observed. With further slip, the load increased again until failure.

(3) Specimens PS-30-60-70-1 to 3, PS-30-60-90-1 to 3, and PS-30-60-120-1 to 3 displayed parabolic load–slip curves between the wythes. For Specimens PS-30-60-70-1 to 3, and PS-30-60-90-1 to 3, a splitting crack appeared at the intersections of the cruciform ribs. The crack width then increased rapidly, leading to a loss of load-bearing capacity in the connector. For Specimens PS-30-60-120-1 to 3, concrete spalling occurred around the anchorage zone of the connector when the peak load was attained. Subsequently, the shear force-slip curve began its descending branch.

(4) For Specimens PS-30-60-70-1 to 3, PS-30-60-90-1 to 3, and PS-30-60-120-1 to 3, the slip curves measured at the anchored ends of the two connectors on the same wythe were nearly identical. The slip difference between the two ends of the same connector did not exceed 0.5 mm.

4.3. Influence Law

This section analyzes the influence of connector embedment depth, outer wythe thickness, and insulation layer thickness on the shear capacity and shear–slip behavior, with the following results.

4.3.1. Effect of Connector Embedment Depth

Figure 10 illustrates the effect of connector embedment depth on the shear–slip behavior. The two specimen groups had embedment depths of 50 mm and 75 mm, respectively, while all other parameters were identical. The comparison reveals that their load–slip curves nearly coincided initially. However, the characteristic load of the specimens with a 50 mm embedment depth was approximately 15.9% lower than that for those with a 75 mm embedment depth. Beyond the characteristic load, the curves diverged significantly. The specimens with a 75 mm embedment depth exhibited a continued increase in load-bearing capacity, whereas those with a 50 mm embedment depth showed minimal additional load resistance, resulting in a final peak load roughly half that of the former group.

4.3.2. Effect of Outer Wythe Thickness

A comparison of load–slip behavior of specimens with different outer wythe thicknesses is presented in Figure 11. The two groups featured 150 mm thick and 200 mm thick outer wythes, respectively, while all other parameters remained identical. The results show that the load–slip curves for the two groups nearly overlapped. The peak load of the specimens with a 200 mm thick outer wythe was 5.0% higher than that of the specimens with a 150 mm thick outer wythe. After accounting for the influence of the self-weight of the wythe, it is evident that the outer wythe thickness had no significant effect on the shear strength or stiffness of the connectors.

4.3.3. Effect of Insulation Layer Thickness

Figure 12 contrasts the load–slip curves obtained from specimens with varying insulation thicknesses. The three specimen groups had insulation thicknesses of 70 mm, 90 mm, and 120 mm, respectively, while all other parameters were kept constant. The results indicates that both the shear stiffness and shear capacity of the connectors decreased with increasing insulation thickness. Specifically, the peak loads for the specimens with 90 mm and 120 mm thick insulation were 26.2% and 36.2% lower, respectively, than that of the specimen with a 70 mm thick insulation layer.

5. Calculation Method for the Shear Capacity of Connectors

5.1. Connector Fracture

FRP is a kind of anisotropic material whose strength criteria differ from those of traditional isotropic materials. In the field of composite materials, commonly used failure theories primarily include the Tsai–Hill criterion, Hoffman criterion, Tsai–Wu criterion, and Hashin criterion. The Tsai–Hill criterion, developed by modifying the von Mises criterion for isotropic materials to apply to anisotropic materials, is simple in form and convenient for calculation. However, it assumes equal material strength in tension and compression, which often contradicts the actual behavior of most composite materials. The Hoffman criterion improves upon the Tsai–Hill criterion by incorporating a linear term to distinguish between tensile and compressive strengths, but it still cannot identify specific failure modes. The Tsai–Wu criterion, an empirical rule based on tensor theory, generally offers high accuracy but requires extensive experimental data to determine its coefficients. The Hashin criterion can predict four failure modes: fiber tensile failure, fiber compressive failure, matrix tensile failure, and matrix compressive failure, making it the most widely used failure criterion for composite materials today.

Analysis of the specimen failure modes and strain data indicates that the connectors predominantly failed in a combined tensile-shear mode. Therefore, the fiber tensile failure mode of the Hashin criterion [25], which accounts for the influence of shear effects, was selected to analyze the shear capacity of the connectors. Under a two-dimensional stress state, the expression for this fiber tensile failure mode is as follows:

$${\displaystyle \frac{{\sigma }_1^2}{S_{11}^2}}+{\displaystyle \frac{{\tau }_{12}^2}{S_{12}^2}}=1.0$$

(1)

where σ₁ is the normal stress at the critical cross-section, τ₁₂ is the shear stress at the critical cross-section, S₁₁ is the tensile strength, and S₁₂ is the shear strength.

The connector is embedded in concrete over a certain length when transmitting shear force within PCSPs, which restricts rotation at its ends. Therefore, the boundary condition can be assumed as fixed at one end and free to slide at the other (Figure 13). Based on this assumption, the various stress components on the connector cross-section under a shear force V_u are as follows:

$$\left\{\begin{array}{l}{\sigma }_1={\displaystyle \frac{V_{\mathrm{u}}ly}{2I}}\\ {\tau }_{12}={\displaystyle \frac{V_{\mathrm{u}}{S}^{\ast }}{Ib}}\end{array}\right.$$

(2)

where V_u represents the shear capacity, l denotes the insulation thickness, I is the moment of inertia, y represents the distance to the neutral axis at the calculation point, S* denotes the static moment of area (of the portion above/below the calculation point level) about the neutral axis, and b is the width of the cross-section at the calculation point.

For the cross-shaped FRP rod connector investigated herein, the cross-sectional stress distribution is shown in Figure 14. Since interlaminar tearing observed in the tests typically initiates at the intersections of the cruciform ribs; therefore, this location was chosen as the critical point for calculation. Based on this selection, the connector shear capacity is determined by:

$$\left\{\begin{array}{c}{\displaystyle \frac{{\sigma }_1^2}{S_{11}^2}}+{\displaystyle \frac{{\tau }_{12}^2}{S_{12}^2}}<1.0\\ {\sigma }_1={\displaystyle \frac{V_{\mathrm{u}}l(D-t)}{4I}}\\ {\tau }_{12}={\displaystyle \frac{V_{\mathrm{u}}(D^2-t^2)}{8I}}\end{array}\right.\Rightarrow V_{\mathrm{u}}={\left[{\displaystyle \frac{1}{\frac{l{(D-t)}^2}{16I^2{S}_{11}^2}+\frac{{(D+t)}^2{(D-t)}^2}{64I^2{S}_{12}^2}}}\right]}^{0.5}$$

(3)

where D represents the outer diameter of the connector, while t denotes the thickness of the flange.

5.2. Concrete Anchorage Failure

The shear capacity corresponding to concrete anchorage failure in the push-out tests can be estimated with reference to the calculation method for anchors exhibiting concrete cone breakout under shear loading.

According to the American standard ACI 349-97 [26], when an anchor fails in tension by pull-out, the concrete failure surface is idealized as a right circular cone with a 45° inclination relative to the concrete surface. The pull-out capacity is determined by the resultant tensile force acting over the horizontally projected area of the breakout cone, as illustrated in Figure 15. For a single anchor not affected by edge distance, anchor spacing, or the thickness of the concrete member, its pull-out capacity is expressed as:

$$N_{\mathrm{u}}=4\pi \sqrt{f_{\mathrm{c}}^{\prime }}{L}_{\mathrm{d}}\left(L_{\mathrm{d}}+D_{\mathrm{b}}\right)$$

(4)

where L_d represents the embedment length of the anchor, while D_b denotes the diameter of the anchor.

Beginning in 1988, Eligehausen, Fuchs, et al. conducted experimental and theoretical research on anchors embedded in concrete under tensile or shear loads. They proposed a calculation method for the pull-out load of anchors, known as the Concrete Capacity Design (CCD) method [28,29,30,31]. The CCD method idealizes the anchor failure surface as a quadratic pyramid, with a 35° angle between the failure surface and the concrete face. Based on this approach, ACI 318-2014 [32] provides a formula for calculating the pull-out capacity of anchors:

$$N_{\mathrm{u}}=24{\lambda }_{\mathrm{a}}\sqrt{f_{\mathrm{c}}^{\prime }}({h_{\mathrm{ef}}}^{1.5})(h_{\mathrm{ef}}\le 11 \mathrm{in}.)$$

(5)

where h_ef represents the effective embedment length of the anchor, and λ_a is a modification factor related to the lightweight concrete.

Regarding the concrete cone breakout capacity of anchors under shear loading, Fuchs et al. suggested that when the anchor embedment depth is less than 2.5 in (63.5 mm), the capacity should range from 1 to 2 times the tensile pull-out capacity. ACI 318-14 [32] specifies that the concrete cone breakout capacity under shear is taken as 1.0 times the tensile pull-out capacity for embedment depths less than 2.5 in, or 2.0 times for embedment depths greater than or equal to 2.5 in.

Based on preliminary test observations, when FRP connectors fail by cone pull-out under shear, the failure cone more closely resembles a circular cone, but with a breakout angle significantly smaller than 45°, approximately 30–35°. Accordingly, by modifying the cone angle in the anchor pull-out capacity formula from ACI 349-97 [27] and introducing a reduction factor for the bending effect of the connector within the insulation layer, an expression for the concrete cone breakout capacity of the connector under shear is proposed as follows:

$$\left\{\begin{array}{l}{V}_{\mathrm{u}}={\alpha }_l{f}_{\mathrm{t}}{A}_{\mathrm{se}}\\ A_{\mathrm{se}}=\pi \left[{\left({\displaystyle \frac{h_{\mathrm{ef}}}{\tan \theta }}+{\displaystyle \frac{D}{2}}\right)}^2-{\left({\displaystyle \frac{D}{2}}\right)}^2\right]\end{array}\right.\Rightarrow V_{\mathrm{u}}={\alpha }_l\pi f_{\mathrm{t}}{\displaystyle \frac{h_{\mathrm{ef}}}{\tan \theta }}\left({\displaystyle \frac{h_{\mathrm{ef}}}{\tan \theta }}+D\right)$$

(6)

where α_l represents the bending effect factor for the connector, determined from experimental results, is taken as 0.158–0.0006l, and θ denotes the breakout cone angle, which is taken as 32° based on test data.

5.3. Shear Capacity Model

The shear capacity of the connector shall be taken as the lesser of the calculated values from the two failure modes:

$$V_{\mathrm{u}}=\min \left\{{\left[{\displaystyle \frac{1}{\frac{l{(D-t)}^2}{16I^2{S}_{11}^2}+\frac{{(D+t)}^2{(D-t)}^2}{64I^2{S}_{12}^2}}}\right]}^{0.5},V_u={\alpha }_l\pi f_{\mathrm{t}}{\displaystyle \frac{h_{\mathrm{ef}}}{\tan \theta }}\left({\displaystyle \frac{h_{\mathrm{ef}}}{\tan \theta }}+D\right)\right\}$$

(7)

Using the method described above, the connector shear capacity in both the present study and reference [33] was calculated. The calculated values were then evaluated against the experimental values, and the comparison results are presented in

Table 7. Calculated and measured shear capacity of specimens.

Specimen No.	Failure Mode	Measured Capacity Vu,test (kN)	Calculated Capacity Vu,cal (kN)	Vu,cal/Vu,test
PS-50-150-30-1~3	Connector fracture	11.93	12.20	1.02
PS-75-150-30-1~3	Connector fracture	14.19	12.20	0.86
PS-75-200-30-1~3	Connector fracture	13.51	12.20	0.90
PS-30-60-70-1~3	Connector fracture	7.04	7.21	1.02
PS-30-60-90-1~3	Connector fracture	6.57	6.87	1.04
PS-30-60-120-1~3	Concrete anchorage failure	5.63	5.26	0.93
CP-S1~CP-S3 [33]	Concrete anchorage failure	3.60	3.39	0.89
CP-S4~CP-S6 [33]	Concrete anchorage failure	2.88	3.05	0.97
Average				0.97
Standard deviation				0.06

. As shown, the mean ratio of calculated to experimental values is 0.97, with a standard deviation of 0.06, indicating good agreement between the predicted and measured shear capacities.

6. Shear–Slip Model

The load–slip curves provide a comprehensive representation of both the shear capacity and slip behavior of connectors, serving as critical experimental evidence for understanding the slip distribution between the wythes of PCSPs. Based on the test results presented in Section 4, it was observed that for some push-out specimens, the load–slip curves entered a plateau stage after reaching the characteristic load. Since the shear stiffness in this plateau stage is zero or negative, the connectors can be considered to have failed. Therefore, the load–slip curve model discussed in this section focuses only on the portion of the curve prior to the connector reaching its shear capacity.

The inter-wythe slip of a PCSP can be divided into two components: slip arising from insufficient anchorage of the connector in the concrete and slip resulting from bending–shear deformation of the connector within the insulation layer. The calculation models for these two slip components are discussed separately in the following sections.

6.1. Slip at the Anchored End of Connector

By summing the slip at the anchored end on either side of the connector -effectively isolating the interface slip by removing the connector deformation component from the total inter-wythe slip-the resulting slip is analogous to the interface slip between the steel beam and concrete slab in steel–concrete composite beams. Following the methodology outlined in references [34], a theoretical model for the slip at the anchored end of the connector is derived through regression analysis of the experimental data:

$${\displaystyle \frac{V}{V_u}}={\displaystyle \frac{s_1}{0.4+0.87s_1}}$$

(8)

where s₁ is the slip at the anchored end of the connector.

The slip at the anchored end of the connector was calculated using the above model and compared with the measured slip. Figure 16 presents the comparison, plotting the shear force on a single connector on the vertical axis. As shown, the theoretical model exhibits good agreement with the experimental results.

6.2. Slip Due to Connector Deformation

The connector within the insulation layer experiences combined bending and shear deformation under the actions at its ends. The theoretical model for the slip resulting from these deformations is expressed as follows:

$$s_2={\displaystyle \frac{V{l}^2}{12EI}}+{\displaystyle \frac{kVl}{GA}}$$

(9)

where s₂ is the slip due to connector deformation; k represents the shear stress non-uniform distribution factor, denoted as $k={\displaystyle \frac{A}{I^2}}{\displaystyle \underset{A}{\int }\frac{S^2}{b^2}}dA$, is taken as 1.2 for rectangular sections. For I-shaped sections, $k\approx A/A_1$ (where A₁ is the web area). For the cruciform section in this study, k = 1.388; E is the elasticity modulus of FRP; and G is the shearing modulus of FRP.

6.3. Shear–Slip Model and Its Experimental Verification

By combining the two slip components described above, the model for the slip between the inner and outer concrete wythes can be expressed as:

$$s={\displaystyle \frac{0.4V}{V_u-0.87V}}+{\displaystyle \frac{V{l}^2}{12EI}}+{\displaystyle \frac{kVl}{GA}}$$

(10)

where s is the inter-wythe slip.

The slip between the inner and outer wythes was calculated using the aforementioned model and compared with the measured slip. The comparison results are presented in Figure 17. The Pearson correlation coefficients (r) for the six sets of the predicted and measured shear–slip curves are listed in

Table 8. The correlation coefficients between the predicted and measured shear–slip curves.

Specimen No.	Vu,cal/Vu,test
PS-50-150-30-1	0.9969
PS-50-150-30-2	0.9965
PS-50-150-30-3	0.9908
PS-75-150-30-1	0.9981
PS-75-150-30-2	0.9719
PS-75-150-30-3	0.9930
PS-75-200-30-1	0.9954
PS-75-200-30-2	0.9899
PS-75-200-30-3	0.9967
PS-30-60-70-1	0.9952
PS-30-60-70-2	0.9956
PS-30-60-70-3	0.9883
PS-30-60-90-1	0.9928
PS-30-60-90-2	0.9915
PS-30-60-90-3	0.9945
PS-30-60-120-1	0.9888
PS-30-60-120-2	0.9870
PS-30-60-120-3	0.9812

. As shown in Table 8, the correlation coefficients between the predicted and measured load–slip curves for all specimens are greater than 0.98, indicating a high consistency in curve shape and variation trend. Compared with the first five specimens, although the correlation coefficient of the sixth specimen is still greater than 0.98, it is slightly lower than that of the previous five. This is mainly because anchorage failure occurred during the test, leading to a reduction in bearing capacity in the later stage, which caused some deviation from the trend of the predicted curve.

7. Discussion

This paper proposes a novel FRP rod connector with a cruciform cross-section, which offers advantages including superior thermal efficiency, excellent mechanical performance, high durability and easy installation, showing promising application prospects in PCSPs. Push-out tests were conducted to investigate the influence of connector embedment depth, outer wythe thickness, and insulation layer thickness on the shear performance of the connectors. Both failure modes, including connector fracture and concrete anchorage failure, were observed. Based on the analysis of how various test parameters affect shear capacity, the following embedment depths are recommended: 75 mm for an outer wythe thickness of 200 mm, 50 mm for 150 mm, and 30 mm for 60 mm. When the insulation layer thickness exceeds 120 mm, concrete anchorage failure occurs, indicating that the FRP material is not fully utilized. For such cases, it is advisable to consider adding anchorage reinforcement or adopting other enhanced anchorage measures.

Based on the Hashin failure criterion, a calculation model for the shear capacity at connector fracture was proposed. Based on the Concrete Capacity Design (CCD) method, the angle of the concrete failure cone was revised in conjunction with experimental results, and a calculation model for the shear capacity at concrete anchorage failure was proposed. The average ratio of the calculated values from the aforementioned models to the experimental values is 0.97, with a standard deviation of 0.06, indicating good agreement. Furthermore, this study decomposes the slip between the inner and outer concrete wythes into two components: slip at the anchored end of the connector and slip due to connector deformation. A predictive model for the shear force-slip curve was subsequently proposed. The correlation coefficients between the predicted and measured load–slip curves for all specimens are greater than 0.98. It should be noted that the aforementioned models were developed for specimens with insulation layer thicknesses ranging from 30 to 120 mm. When the insulation layer thickness exceeds 120 mm, the configuration of the concrete failure cone may change. Consequently, the applicability of these models requires further verification.

8. Conclusions

This study developed a novel cross-shaped FRP rod connector for PCSPs and conducted both experimental and theoretical investigations on its shear performance. Through shear tests on 18 connectors, the failure modes, shear–slip behavior, and shear capacity of the connectors were examined, leading to the following conclusions:

(1) Two primary failure modes were observed: fracture of the connector and concrete anchorage failure. Specimens with insulation layer thicknesses ranging from 30 mm to 90 mm exhibited connector fracture, while those with a 120 mm thick insulation layer experienced concrete anchorage failure.

(2) For 30 mm thick insulation specimens, the stiffness dropped sharply after reaching a certain load, and the load–slip curve entered a plateau stage. As slip increased further, the load began to rise again until the connector failed in shear. For specimens with insulation thicknesses of 70 mm, 90 mm, and 120 mm, the load–slip curves all exhibited a parabolic shape, with failure occurring shortly after the peak load.

(3) For an insulation thickness of 30 mm, the load–slip curves of specimens with 50 mm and 75 mm embedment depths were nearly identical initially. However, beyond the characteristic load, the curves diverged significantly. Specimens with a 75 mm embedment depth showed a continued increase in load-bearing capacity, whereas specimens with a 50 mm embedment depth exhibited minimal additional load, resulting in a final peak load approximately half that of the former group.

(4) Both the shear stiffness and shear resistance of the connectors decreased with increasing insulation layer thickness. Specifically, the peak loads of specimens with 90 mm and 120 mm insulation were 26.2% and 36.2% lower, respectively, than that of the specimen with a 70 mm insulation layer.

Based on these findings, the underlying mechanisms for the observed failure modes were analyzed. A calculation model for the shear capacity corresponding to connector fracture was proposed based on the Hashin criterion for composite materials. Additionally, a formula for the shear capacity associated with concrete anchorage failure was derived using a cone pull-out model. The calculated results show good agreement with the experimental data. Furthermore, a theoretical load–slip model for the cross-shaped FRP rod connector was established.

The findings of this paper could provide support for the design and application of the cross-shaped FRP rod connectors in PCSPs. Future studies would focus more on the shear performance and detailing optimization of this connector when applied to PCSPs with thicker insulation layers, aiming to facilitate its application in ultra-low energy consumption buildings.